Predictor-based control of linear systems with large and variable measurement delays

Abstract

This paper concerns the problem of the control of linear systems by means of feedback from delayed output, where the delay is known and time-varying. The main advantage of the approach is that it can be applied to systems with any delay bound, i.e. not only small delays. The predictor is based on a combination of finite-dimensional elementary predictors whose number can be suitably chosen to compensate any delay. The single-predictor element is an original proposal, and the class of delays to which the schema can be applied includes, but it is not limited to, continuous delay functions.

Delayed boundary control of a heat equation under discrete-time point measurements

“…(2) The observer (2) is obtained by shifting the plant (1) in time by r and introducing a correcting term. The time-varying injection gain Le −αo(t+r−s k ) will allow to guarantee that the observation error decays with the rate α o [21]. The shape functions b i ∈ L 2 (0, 1) are given by where…”

“…Proof: The transformation (16) has an inverse, which is bounded in H 1 norm (see, e.g., [25]). Therefore, there exists a constantC such that ẑ(•, t) H 1≤C w(•, t) H 1 (21) ≤CC w e − min{αo,αc}t z(•, 0) H 1 for t ≥ t 0 . Since z(x, t) =ẑ(x, t − r) −z(x, t), the latter together with (5) imply (22).…”

“…This idea can be used to analyse chain observers [12], [13], [14], [15] and sequential predictors [16], [17], [18], [19], [20]. In [21] a time-varying injection gain was introduced in the observer to improve its exponential convergence under delayed measurements. This result was revisited in [22] to increase the period of a sampled-data system.…”

“…Inspired by the ideas of sequential predictors [16], [17], [18], [19], [20], we construct an observer that estimates the future value of the state using the sampled measurements. By introducing a time-varying injection gain [21] and performing the stability analysis in a manner similar to [34], [37], we show that the observation error exponentially vanishes with any desired decay rate if the delays and sampling are small enough while the number of sensors is large enough. Such observer allows to eliminate the constant input delay.…”

Prediction-based control of LTI systems with input and output time-varying delays

“…In practice, this is generally not possible to know the delay value in advance that is why alternative predictive techniques have been developed. In [20], the sub predictor method developed in [21] has been modified and extended to time-varying output and input delays. The advantage of this method is that it is finite dimensional.…”